Do you sculpt a character for a game with the clothing as one object or separate
Well, you could do it either way. However the usual approach is to have any changeable clothing be separate..

Often you wouldn't sculpt the clothing any way, because the shape of the mesh isn't important and will get deformed instead by the things under it using a simulation. As I said in my comment it's a bit complicated, so you really need to learn more about the cloth simulation system.

If you didn't want to use the simulation, either because you have very rigid clothes/armour or it's super tight against the character then I would probably model and retopo the character first plus rig it. For tight stuff, just duplicate and lift the retopoed surface from the relevant part of your character. Since it's tight fitting you can use the same bone weights for the most part and save a bunch of time.

But it all depends on what you need. For realistic clothes and armour it's best to model the body and then put the bits on of top of it. If you don't care then you could skip making any parts of the body that are fully covered.

If for example your character appears in three different sets of full plate/powered armour then you're probably better off just making three entire characters that just share a head or something. They may even need different animations if the clothing is sufficiently different.

There isn't really one right or wrong way. It's just having a base body that you can put different clothes on is more flexible and extendible, especially if they can share one animation set.